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arXiv:1812.06726 · Nuclear Physics B

Machine Learning as a universal tool for quantitative investigations of phase transitions

Cinzia Giannetti, Biagio Lucini A, Davide Vadacchino

Published May 16, 2019• 51 citations
arXivHTMLPDF

Abstract

The problem of identifying the phase of a given system for a certain value of the temperature can be reformulated as a classification problem in Machine Learning. Taking as a prototype the Ising model and using the Support Vector Machine as a tool to classify Monte Carlo generated configurations, we show that the critical region of the system can be clearly identified and the symmetry that drives the transition can be reconstructed from the performance of the learning process. The role of the discrete symmetry of the system in obtaining this result is discussed. A finite size analysis of the learned Support Vector Machine decision function allows us to determine the critical temperature and critical exponents with a precision that is comparable to that of the most efficient numerical approaches relying on a known Hamiltonian description of the system. For the determination of the critical temperature and of the critical exponent connected with the divergence of the correlation length, other than the availability of a range of temperatures having information on both phases, the method we propose does not rest on any physical input on the system, and in particular is agnostic to its Hamiltonian, its symmetry properties and its order parameter. Hence, our investigation provides a first significant step in the direction of devising robust tools for quantitative analyses of phase transitions in cases in which an order parameter is not known.

DOI
10.1016/j.nuclphysb.2019.114639

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The problem of identifying the phase of a given system for a certain value of the temperature can be reformulated as a classification problem in Machine Learning. Taking as a prototype the Ising model and using the Support Vector Machine as a tool to classify Monte Carlo generated configurations, we show that the critical region of the system can be clearly identified and the symmetry that drives the transition can be reconstructed from the performance of the learning process. The role of the discrete symmetry of the system in obtaining this result is discussed. A finite size analysis of the learned Support Vector Machine decision function allows us to determine the critical temperature and critical exponents with a precision that is comparable to that of the most efficient numerical approaches relying on a known Hamiltonian description of the system. For the determination of the critical temperature and of the critical exponent connected with the divergence of the correlation length, other than the availability of a range of temperatures having information on both phases, the method we propose does not rest on any physical input on the system, and in particular is agnostic to its Hamiltonian, its symmetry properties and its order parameter. Hence, our investigation provides a first significant step in the direction of devising robust tools for quantitative analyses of phase transitions in cases in which an order parameter is not known.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena.We train a restricted Boltzmann machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods.From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system.We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near-criticality region of the Ising model.In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the renormalization group (RG) flow of the lattice model.Our results suggest an explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points.Then from the reconstructed data we deduce the critical exponent associated with the magnetization to find satisfactory agreement with the actual physical value.We assume no prior knowledge about the criticality of the system and its Hamiltonian.
In recent years Machine Learning has proved to be successful in many technological applications and scientific tasks, such as image and speech recognition, natural language understanding and more. In this work we apply both supervised and unsupervised Machine Learning to three key problems in statistical physics: design of Hamiltonian from data, phase recognition and study of critical properties of a system undergoing phase transition. As an example of model design, we use a dataset with spin configurations and corresponding energies randomly sampled from a one dimensional Ising model. We then make two guesses for the correct Hamiltonian: the former involving interactions of spins with an external local field, the latter involving two body interactions among spins. The coupling constants are determined with a linear penalized regression, comparing the effects of L1 and L2 penalization terms of the cost function. We pay specific attention to the problem of overfitting and to the validation process, which is critical for accepting or rejecting the proposed model. The phase recognition problem is faced with the two dimensional Ising and XY models as examples. After showing the limits of a simple softmax regression for this task, we build suitable neural networks to overcome these limits. In particular, a feed forward network is built and the learning process is investigated for the Ising model; while a more sophisticated convolutional network is proposed for the XY model in order to detect local topological structures. The last part of the work is dedicated to the unsupervised study of phase transitions, and the determination of critical properties (order parameter, critical temperature, critical exponents). The discussed techniques are Principal Component Analysis (PCA) for dimensional reduction and K-means clustering for organizing data into subsets with specific properties. PCA is applied to the two dimensional Ising, Potts and XY models, and it is used to find suitable order parameters. The study of the proposed order parameter as a function of temperature provides evidence of phase transition, and a finite size scaling allows to extrapolate both the critical temperature in the thermodynamic limit, and the ν\nuν critical exponent for the correlation length. K-means clustering is applied to equilibrium configurations of the two dimensional Ising model, before and after the dimensional reduction, and the critical temperature is estimated. Moreover, a clustering of relaxation curves of magnetization in a Monte Carlo dynamics is used to build a phase diagram on the parameter space of temperature and magnetic field. A synergy of C++, Python and Wolfram Mathematica 12.0 is used throughout this work in order to sample input datasets and to build and control customized neural networks and learning tools. The most relevant codes are provided in the appendix.
Detection of phase transitions is a critical task in statistical physics, traditionally pursued through analytic methods and direct numerical simulations. Recently, machine-learning techniques have emerged as promising tools in this context, with a particular focus on supervised and unsupervised learning methods, along with nonlearning approaches. In this paper, we study the performance of unsupervised learning in detecting phase transitions in the J_{1}-J_{2} Ising model on the square lattice. The model is chosen due to its simplicity and complexity, thus providing an understanding of the application of machine-learning techniques in both straightforward and challenging scenarios. We propose a simple method based on a direct comparison of configurations. The reconstruction error, defined as the mean-squared distance between two configurations, is used to determine the critical temperatures. The results from the comparison of configurations are contrasted with those of the configurations generated by variational autoencoders. Our findings highlight that for certain systems a simpler method can yield results comparable to more complex neural networks. This paper contributes to the broader understanding of machine-learning applications in statistical physics and introduces an efficient approach to the detection of phase transitions using machine determination techniques.
We investigate phase transitions in the two-dimensional Ising model with competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions, where the NNN coupling generates a degenerate superantiferromagnetic (SAF) ground state, by combining machine-learning methods with Monte Carlo simulations and finite-size scaling (FSS). Going beyond earlier studies, we show that unsupervised learning can recover order-parameter and Binder cumulant analogs that obey FSS. Principal component analysis (PCA) reveals the two-component structure of the SAF order parameter, and k-means clustering is used to visualize qualitative differences between ordered and disordered configurations in the principal-component space. A fully connected neural network (FCNN), trained without prior knowledge of the order parameter or the critical temperature, learns the strength of SAF stripe ordering and uses it as the internal feature for phase classification. FSS of thermodynamic observables, the PCA-derived quantity, and the FCNN output yields consistent estimates of the critical temperature and correlation-length exponent. These results demonstrate that machine learning can decode ordering patterns, extract meaningful order-parameter analogs, and reproduce critical behavior in frustrated Ising systems.
We propose two machine-learning methods based on neural networks, which we respectively call the phase-classification method and the temperature-identification method, for detecting different types of phase transitions in the XXZ models without prior knowledge of their critical temperatures. The XXZ models have exchange couplings which are anisotropic in the spin space where the strength is represented by a parameter $\mathrm{\ensuremath{\Delta}}(>0)$. The models exhibit second-order phase transition when $\mathrm{\ensuremath{\Delta}}>1$, but the Berezinskii-Kosterlitz-Thouless (BKT) phase transition when $\mathrm{\ensuremath{\Delta}}<1$. In the phase-classification method, the neural network is trained using spin or vortex configurations of well-known classical spin models other than the XXZ models, e.g., the Ising models and the XY models, to classify those of the XXZ models to corresponding phases. We demonstrate that the trained neural network successfully detects the phase transitions for both $\mathrm{\ensuremath{\Delta}}>1$ and $\mathrm{\ensuremath{\Delta}}<1$, and the evaluated critical temperatures coincide well with those evaluated by conventional numerical calculations. In the temperature-identification method, on the other hand, the neural network is trained so as to identify temperatures at which the input spin or vortex configurations are generated by the Monte Carlo thermalization. The critical temperatures are evaluated by analyzing the optimized weight matrix, which coincides with the result of numerical calculation for the second-order phase transition in the Ising-like XXZ model with $\mathrm{\ensuremath{\Delta}}=1.05$ but cannot be determined uniquely for the BKT transition in the XY-like XXZ model with $\mathrm{\ensuremath{\Delta}}=0.95$.
We present a physical interpretation of machine learning functions, opening up the possibility to control properties of statistical systems via the inclusion of these functions in Hamiltonians.In particular, we include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system.Results in the two-dimensional Ising model evidence that the field can induce an order-disorder phase transition by breaking or restoring the symmetry, in contrast with the field of the conventional order parameter which causes explicit symmetry breaking.The critical behavior is then studied by proposing a Hamiltonian-agnostic reweighting approach and forming a renormalization group mapping on quantities derived from the neural network.Accurate estimates of the critical point and of the critical exponents related to the operators that govern the divergence of the correlation length are provided.We conclude by discussing how the method provides an essential step toward bridging machine learning and physics.